CONSTRUCTION OF THE HILBERT CLASS FIELD OF SOME IMAGINARY QUADRATIC FIELDS. Jangheon Oh

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1 Korean J. Math. 26 (2018), No. 2, pp CONSTRUCTION OF THE HILBERT CLASS FIELD OF SOME IMAGINARY QUADRATIC FIELDS Jangheon Oh Abstract. In the paper [4], we constructed 3-part of the Hilbert class field of imaginary quadratic fields whose class number is divisible exactly by 3. In this paper, we extend the result for any odd prime p. 1. Introduction When the order of the sylow 3-subgroup of the ideal class group of an imaginary quadratic field k = Q( d) and Q( 3d) is 3 and 1 respectively, we [4] explicitly constructed 3-part of the Hilbert class field of k. We briefly explain the construction. First, using Kummer theory, we construct everywhere unramified extension H z = k z (α) over k z = k(ζ 3 ) such that the degree [H z : k z ] is 3. The Galois group of H z /k is Z 6 and the unique subfield M of H z, whose degree over k is 3, is the desired 3-part of Hilbert class field of k. Moreover, M is k(β), where β = T r Hz/M(α) and α is a unit of Q( 3d). The explicit computation of α is given in the paper [3]. In this paper, we extend the result for any odd prime p. The proof in this paper is similar to that in the case of p = 3. Throughout this paper, d is a square free positive integer and k an imaginary quadratic field Q( d) such that k Q(ζ p ) = Q. Received March 08, Revised June 5, Accepted June 7, Mathematics Subject Classification: 11R23. Key words and phrases: Iwasawa theory,hilbert class field, Kummer extension. c The Kangwon-Kyungki Mathematical Society, This is an Open Access article distributed under the terms of the Creative commons Attribution Non-Commercial License ( -nc/3.0/) which permits unrestricted non-commercial use, distribution and reproduction in any medium, provided the original work is properly cited.

2 294 Jangheon Oh 2. Proof of Theorems Denote k(ζ p ) by L. Then L is a CM field, and let L + be the maximum real subfield of L. Proposition 1. Let p be an odd prime. Then L + = Q( d sin( 2π p ) + cos(2π p )). Proof. Denote d, ζ p ζ p 1, ζ p + ζ p 1 by α, β, γ respectively. Note that αβ and γ are real numbers and L = Q(α, β, γ) = Q(αβ, γ)(α). Hence L + = Q(αβ, γ) = Q( d sin( 2π), cos( 2π )). Let σ be an element p p of Gal(L/Q) such that σ(α) = α and σ(ζ p ) = ζ p. Then σ(αβ + γ) = αβ + γ Q(αβ + γ), which completes the proof. Remark 1. Note that L + = Q( 3d) when p = 3. We denote by M K, A K, E K, χ, ω the maximal unramified elementary abelian p-extension of a number field K, the p-part of ideal class group of K, the set of units of K, the nontrivial character of Gal(k/Q), and the Teichmuller character, respectively. By Kummer theory, there is a subgroup B L /(L ) p such that M L = L( p B) and a nondegenerate pairing Gal(M L /L) B µ p. Since Gal(M L /L) A L /A L p, we have a map φ : B A L,p := {x A L x p = 1} and Ker(φ) subgroup of E L /E L p. From this pairing, we have an induced nondegenerate pairing Gal(M L /L) χ B χω µ p, where we write M = ψ M ψ for the character ψ s of G = Gal(L/Q) and the Z p [G]-module M(See [5]). The map φ is G-linear, so we have an induced map φ χω from φ φ χω : B χω (A L,p ) χω.

3 Construction of the Hilbert class field of some imaginary quadratic fields 295 Note that (E L /E L p ) χω = (E L +/E L + p ) χω and the order of (E L +/E L + p ) χω is p(see [2]). Hence we have p-rank(gal(m L /L) χ ) = p-rank(b χω ) p-rank(ker(φ χω )) + p-rank((a L /A L p ) χω ) 1 + p-rank((gal(m L /L) χω ) Since [E L : µ p E L +] = 1 or 2 and χ( ω) is an odd character, the order of (E L /E L p ) χ is 1. So similarly as above we have p-rank(gal(m L /L) χω ) = p-rank(b χ ) p-rank(ker(φ χ )) + p-rank((a L /A L p ) χ ) p-rank(gal(m L /L) χ ) Since p and p 1 is relatively prime, we see that Gal(M L /L) χ Gal(M k /k) χ A k /A p k. Therefore we proved the following theorem. Theorem 2.1. We have the inequality. p-rank((a L /A L p ) χω ) p-rank(a k /A k p ) 1 + p-rank((a L /A L p ) χω ). Remark 2. Theorem 2.1 is already known for p = 3. The above proof just follows the proof for p = 3. Let N K be the maximal abelian p-extension of a number field K unramified outside above p, and X K be Gal(N K /K)/Gal(N K /K) p. Then, by Kummer theory again, we have a nondegenerate pairing S χω X L,χ µ p, where S is a subset of L /L p corresponding to X L. It is seen [2] that S E L /E L p A L /A L p < p > / < p > p. So S χω = (E L /E p L ) χω (A L /A p L ) χω. Note again that the order of (E L /E p L ) χω is p. Hence, if the order of (A L /A p L ) χω is 1, then (N L ) χ = L( p ɛ), where ɛ (E L +/E p L + ) χω. Theorem 2.2. Let p be a prime p > 3. Assume that the order of A k is p and that of (A L /A L p ) χω is 1. Then M k is the unique subfield of L( p ɛ) such that the degree [M k : k] = p, where ɛ (E L +/E L + p ) χω. Moreover, M k = k(t r (NL ) χ/m k ( p ɛ))

4 296 Jangheon Oh Proof. Since p and p 1 is relatively prime, we see that (X k ) χ (X L ) χ. The complex conjugate acts on the Hilbert class field of k inversely, so the condition in Theorem 2.2 implies that M k = (M k ) χ = (N k ) χ. The galois group Gal((M L ) χ /k) is an abelian group of order p(p 1), so (M L ) χ contains the unique subfield F whose degree over k is p. Hence M k = F and by Kummer theory(see for example [1]) we see that F = k(t r (NL ) χ/m k ( p ɛ)). Remark 3. When the order of A k is p, then that of (A L /A L p ) χω is 1or p by Theorem 2.1. We proved the above theorem for p = 3(See [4]). The construction of the unit ɛ in Theorem 2.2 is given in [3]. The compositum L k of all Z p -extension of k is the Z 2 p-extension of k. The L k is the product of the cyclotomic Z p -extension and the anticyclotomic Z p -extension of k. The following theorem tells when the first layer k1 a of the anti-cyclotomic Z p -extension is unramified everywhere over k. Theorem 2.3. Let p be a prime p(> 3). The first layer k a 1 of the anticyclotomic Z p -extension is unramified everywhere over k if and only if p-rank(a k /A k p ) = 1 + p-rank((a L /A L p ) χω ). Proof. By class field theory, Gal(N k /H k ) ( p p U 1,p)/E Z 2 p, where H k is the p-part of Hilbert class field of k, U 1,p local units congruent to 1 modulo p,, and E the closure of global units of k in p p U 1,p. And note that (N k ) χ is the compositum of the anti-cylotomic Z p -extension of k and H k. Assume that p-rank(a k /A k p ) = 1 + p-rank((a L /A L p ) χω ). Then since (X k ) χ (X L ) χ, we have p-rank((x k ) χ ) = p-rank((x L ) χ = p-rank(s χω ) = p-rank((e + L /E+ L p ) χω ) + p-rank((a L /A L p ) χω ) = 1 + p-rank((a L /A L p ) χω ) = p-rank(a k /A k p ). Hence the first layer k a 1 should be a part of H k. Assume not that p-rank(a k /A k p ) = 1 + p-rank((a L /A L p ) χω ). Then, by Theorem 2.1, p-rank(a k /A k p ) = p-rank((a L /A L p ) χω ), and hence p-rank((x k ) χ ) = 1 + p-rank(a k /A k p ), so the first layer k a 1 should be ramified over k.

5 Construction of the Hilbert class field of some imaginary quadratic fields 297 References [1] H.Cohen, Advanced Topics in Computational Number Theory, Springer, [2] J.Minardi, Iwasawa modules for Z d p-extensions of algebraic number fields, Ph.D dissertation, University of Washington, [3] J.Oh, On the first layer of anti-cyclotomic Z p -extension of imaginary quadratic fields, Proc. of The Japan Acad. Ser.A 83 (2007) (3), [4] J.Oh, Construction of 3-Hilbert class field of certain imaginary quadratic fields, Proc. of The Japan Acad. Ser.A 86 (2010) (1), [5] L.Washington, Introduction to cyclotomic fields, Springer, New York, Jangheon Oh Faculty of Mathematics and Statistics Sejong University Seoul , Korea oh@sejong.ac.kr